| L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.955 − 1.06i)3-s + (−0.978 + 0.207i)4-s + (0.663 − 2.13i)5-s + (0.955 − 1.06i)6-s − 1.22·7-s + (−0.309 − 0.951i)8-s + (0.100 − 0.955i)9-s + (2.19 + 0.436i)10-s + (−2.40 + 1.75i)11-s + (1.15 + 0.839i)12-s + (0.313 − 2.97i)13-s + (−0.127 − 1.21i)14-s + (−2.90 + 1.33i)15-s + (0.913 − 0.406i)16-s + (0.501 + 0.106i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.551 − 0.612i)3-s + (−0.489 + 0.103i)4-s + (0.296 − 0.954i)5-s + (0.390 − 0.433i)6-s − 0.462·7-s + (−0.109 − 0.336i)8-s + (0.0334 − 0.318i)9-s + (0.693 + 0.138i)10-s + (−0.726 + 0.527i)11-s + (0.333 + 0.242i)12-s + (0.0868 − 0.826i)13-s + (−0.0342 − 0.325i)14-s + (−0.748 + 0.344i)15-s + (0.228 − 0.101i)16-s + (0.121 + 0.0258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0383037 - 0.292488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0383037 - 0.292488i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.663 + 2.13i)T \) |
| 19 | \( 1 + (-0.00483 - 4.35i)T \) |
| good | 3 | \( 1 + (0.955 + 1.06i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + (2.40 - 1.75i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.313 + 2.97i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.501 - 0.106i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (0.282 + 0.125i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (7.28 - 1.54i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (0.0174 + 0.0535i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.46 - 1.06i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.49 - 1.11i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.211 - 0.367i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.74 - 1.00i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-7.41 + 1.57i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (5.14 - 2.28i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (6.84 + 3.04i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (3.30 - 3.66i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-6.32 - 7.02i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.0646 - 0.615i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (7.87 + 8.75i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (4.64 + 14.2i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (15.1 + 6.75i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (6.39 + 7.09i)T + (-10.1 + 96.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.649309741155247686423102504664, −8.673910928161282844127343373057, −7.83146899169173747517506421092, −7.12571027802349772392811339682, −5.97634907499846082400607070927, −5.64862428672362529943193019151, −4.60729573916753443886159985327, −3.37392643788444852252187016270, −1.59159433567568653374137019594, −0.14018157622159865548020946242,
2.08678141836358664935128290159, 3.08269569456482351722962582947, 4.09171015554850004660417139100, 5.15743497155918058242404171414, 5.94065052271772656966923252615, 6.92684172708466196982663797057, 7.934544732794057292568619430104, 9.172005155591624275492280766116, 9.809376905768672150467565257796, 10.54429347209736297580729603216
